Normal traces and applications to continuity equations on bounded domains

TL;DR

This paper investigates the properties of the normal Lebesgue trace of vector fields, demonstrating its relation to divergence measures and BV functions, and applies these findings to establish uniqueness results for continuity equations on bounded domains.

Contribution

It introduces new properties of the normal Lebesgue trace, including the Gauss-Green identity, and applies these to improve uniqueness results for continuity equations without requiring full boundary BV regularity.

Findings

Normal Lebesgue trace satisfies Gauss-Green identity.

Counterexamples show the trace is between measure-divergence and BV functions.

Results enable removing boundary BV regularity assumption for uniqueness.

Abstract

In this work, we study several properties of the normal Lebesgue trace of vector fields introduced by the second and third author in [22] in the context of the energy conservation for the Euler equations in Onsager-critical classes. Among other things, we prove that the normal Lebesgue trace satisfies the Gauss-Green identity and, by providing explicit counterexamples, that it is a notion sitting strictly between the distributional one for measure-divergence vector fields and the strong one for $BV$ functions. These results are then applied to the study of the uniqueness of weak solutions for continuity equations on bounded domains, allowing to remove the assumption in [19] of global $BV$ regularity up to the boundary, at least around the portion of the boundary where the characteristics exit the domain or are tangent. The proof relies on an explicit renormalization formula completely characterized by the boundary datum and the positive part of the normal Lebesgue trace. In the case when the characteristics enter the domain, a counterexample shows that achieving the normal trace in the Lebesgue sense is not enough to prevent non-uniqueness, and thus a $BV$ assumption seems to be necessary to get uniqueness.